In algebraic geometry , motives or sometimes motifs , following French usage is a theory proposed by Alexander Grothendieck in the 's to unify the vast array of similarly behaved cohomology theories such as singular cohomology , de Rham cohomology , etale cohomology , and crystalline cohomology. Philosophically, a 'motif' is the 'cohomology essence' of a variety.
In that article, a motive is a 'system of realisations'. That is, a tuple. The theory of motives was originally conjectured as an attempt to unify a rapidly multiplying array of cohomology theories, including Betti cohomology , de Rham cohomology , l -adic cohomology , and crystalline cohomology. The general hope is that equations like. From another viewpoint, motives continue the sequence of generalizations from rational functions on varieties to divisors on varieties to Chow groups of varieties.
The generalization happens in more than one direction, since motives can be considered with respect to more types of equivalence than rational equivalence. The admissiable equivalences are given by the definition of an adequate equivalence relation. The category of pure motives often proceeds in three steps. The morphisms are correspondences. In detail, let X and Y be smooth projective varieties and consider a decomposition of X into connected components:.
It is a preadditive category. The sum of morphisms is defined by. The motive [ X ] is often called the motive associated to the variety X. As intended, Chow eff k is a pseudo-abelian category. The direct sum of effective motives is given by. The tensor product of effective motives is defined by. The tensor product of morphisms may also be defined. To proceed to motives, we adjoin to Chow eff k a formal inverse with respect to the tensor product of a motive called the Lefschetz motive.
The effect is that motives become triples instead of pairs. The Lefschetz motive L is. Then we define the category of pure Chow motives by. In order to define an intersection product, cycles must be "movable" so we can intersect them in general position. Choosing a suitable equivalence relation on cycles will guarantee that every pair of cycles has an equivalent pair in general position that we can intersect. The Chow groups are defined using rational equivalence, but other equivalences are possible, and each defines a different sort of motive.
Examples of equivalences, from strongest to weakest, are. The literature occasionally calls every type of pure motive a Chow motive, in which case a motive with respect to algebraic equivalence would be called a Chow motive modulo algebraic equivalence. This should be such that motivic cohomology defined by. The existence of such a category was conjectured by Alexander Beilinson. Instead of constructing such a category, it was proposed by Deligne to first construct a category DM having the properties one expects for the derived category. Getting MM back from DM would then be accomplished by a conjectural motivic t-structure.
Lectures on the Theory of Pure Motives (University Lecture Series)
The current state of the theory is that we do have a suitable category DM. Already this category is useful in applications. Vladimir Voevodsky 's Fields Medal -winning proof of the Milnor conjecture uses these motives as a key ingredient. There are different definitions due to Hanamura, Levine and Voevodsky.
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They are known to be equivalent in most cases and we will give Voevodsky's definition below. The category contains Chow motives as a full subcategory and gives the "right" motivic cohomology. However, Voevodsky also shows that with integral coefficients it does not admit a motivic t-structure. Its elements are called finite correspondences.
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The only non-trivial part of this "definition" is the fact that we need to describe compositions. These are given by a push-pull formula from the theory of Chow rings. If we localize this category with respect to the smallest thick subcategory meaning it is closed under extensions containing morphisms.
The hom-groups are then the colimit. A commonly applied technique in mathematics is to study objects carrying a particular structure by introducing a category whose morphisms preserve this structure. Then one may ask, when are two given objects isomorphic and ask for a "particularly nice" representative in each isomorphism class. The classification of algebraic varieties, i. The relaxed question of studying varieties up to birational isomorphism has led to the field of birational geometry.
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So what are these motives? The idea goes back to the s. If one wants to study an algebraic variety defined, say, by a set of polynomial equations with rational coefficients, one of the best tools is cohomology. But which cohomology?
One of the most important results of the period was a comparison theorem that showed that in fact these theories were very closely related. Motives turned out to be a very fruitful and inspirational idea. Mathematicians working in arithmetic algebraic geometry talk about them all the time. But pinning them down precisely and proving theorems about them has turned out to be quite hard. Grothendieck formulated quite precisely what these motives should look like.
Those are far from settled. The lectures in this book were given by Murre, originally in and then many other times in various settings. With the help of Nagel and Peters, they have been written up completely and can now be made available to all who want to know more. They cover the construction and theory of pure motives, but also offer some hints about how to go beyond them.